Yes, they can. When you send someone's Snap story, there are indicators on Snapchat that can show the sender. For example, if they check the views or interactions related to their story, your name or username might be visible among those who have interacted with it, which includes sending the story. Also, Snapchat has privacy settings but these don't typically hide the fact that you've sent someone's story in the normal course of using the app.
Yes. Snapchat has features that allow the owner of a Snap story to see various interactions with their story. Sending a story is one such interaction. If they go to the analytics or the list of interactions for their story, they will be able to see that you sent it. It's part of the transparency that Snapchat offers to its users to know how their content is being spread and interacted with.